Question: Simplify the following expression: $y = \dfrac{3x^2- 5x- 28}{x - 4}$
Answer: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(3)}{(-28)} &=& -84 \\ {a} + {b} &=& &=& {-5} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-84$ and add them together. Remember, since $-84$ is negative, one of the factors must be negative. The factors that add up to ${-5}$ will be your ${a}$ and ${b}$ When ${a}$ is ${7}$ and ${b}$ is ${-12}$ $ \begin{eqnarray} {ab} &=& ({7})({-12}) &=& -84 \\ {a} + {b} &=& {7} + {-12} &=& -5 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({3}x^2 +{7}x) + ({-12}x {-28}) $ Factor out the common factors: $ x(3x + 7) - 4(3x + 7)$ Now factor out $(3x + 7)$ $ (3x + 7)(x - 4)$ The original expression can therefore be written: $ \dfrac{(3x + 7)(x - 4)}{x - 4}$ We are dividing by $x - 4$ , so $x - 4 \neq 0$ Therefore, $x \neq 4$ This leaves us with $3x + 7; x \neq 4$.